Concerning the linear action of discrete subgroups of SL(2,R) on the plane. Mémoire d’Habilitation à Diriger des Recherches François Maucourant To cite this version: François Maucourant. Concerning the linear action of discrete subgroups of SL(2,R) on the plane. Mémoire d’Habilitation à Diriger des Recherches. Mathematics [math]. Université Rennes 1, 2014. tel-01428855 HAL Id: tel-01428855 https://hal.archives-ouvertes.fr/tel-01428855 Submitted on 11 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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MEMOIRE´ D'HABILITATION pr´esent´eepar A` DIRIGER DES RECHERCHES Fran¸coisMaucourant Universit´ede Rennes 1 pr´epar´ee`al'Institut de recherche Mention : Math´ematiques math´ematiquede Rennes, UMR 6625 du CNRS Ecole´ doctorale Matisse CONCERNING THE LINEAR soutenue le 2 d´ecembre 2014 devant le jury compos´ede : ACTION OF DISCRETE Nalini Anantharaman SUBGROUPS OF SL(2,R) Professeure, Universit´ede Strasbourg / Rapporteure Yves Benoist ON THE PLANE Directeur de Recherche, Universit´eParis-Sud / Examinateur Fran¸coise Dal'bo Professeure, Universit´ede Rennes 1 / Examinatrice Marc Peign´e Professeur, Universit´ede Tours / Examinateur Bertrand R´emy Professeur, Ecole´ Polytechnique / Examinateur au vu des rapports de : Nalini Anantharaman Professeure, Universit´ede Strasbourg Amos Nevo Professeur, Technion Arnaldo Nogueira Professeur, Aix-Marseille Universit´e Contents Remerciements 7 Introduction 9 Chapter I. Growth in semi-simple Lie groups 15 I.1. Asymptotic growth of Haar measure 15 I.2. A procedure to compute the volume growth 16 I.3. An example : G2 as an automorphism group 16 I.4. And for lattices ? 19 I.5. Other related results 24 Chapter II. Distribution of orbits of lattices on the plane 25 II.1. Local distributions results 25 II.2. Some remarks 27 II.3. Some ideas of the proofs 29 Chapter III. Action of finitely generated groups on the plane 37 III.1. Ergodic theory on geometrically finite surfaces 37 III.2. The action on the plane 43 III.3. Large scale α = 1 47 Chapter IV. Random orbits on the plane 51 IV.1. Heuristic and motivations 51 IV.2. Random walk setup 54 + IV.3. The real functions k,k ,kη 55 IV.4. Construction of eigenmeasures 61 IV.5. Green-Poincar´eSeries and (Γ; µ)-conformal measures 64 IV.6. Examples 66 IV.7. Further comments 71 Bibliography 73 3 4 CONTENTS Les intellectuels, ne l'oublions pas, ont bien int´egr´el'id´eequ'il faut que les choses paraissent compliqu´ees.Sinon, `aquoi servent-ils ? 1 Noam Chomsky (1928-) 1 Let's not forget that intellectuals have very well grasped the idea that things have to look complicated. Otherwise, what are they good for ? CONTENTS 5 Publications (1) Approximation diophantienne, dynamique des chambres de Weyl, et r´epartition d'orbites de r´eseaux, PhD Thesis, 2002, Universit´eLille 1.(TEL) (2) Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques, Ergodic Theory and Dyn. Sys. 23 (2003), 193-205. (article) (3) with A. Gorodnik, Proximality and Equidistribution on the Furstenberg bound- ary, Geometriae Dedicata, 113 (2005), p 197-213. (Lien arXiv) (4) Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel Journal of math- ematics, 152 (2006), p 143-155. (article). (5) Homogeneous asymptotic limits of Haar measures of semisimple Lie groups and their lattices, Duke Mathematical Journal, vol 136 (2), 2007, p 357-399. (arXiv). (6) With A. Gorodnik et H. Oh, Manin's and Peyre's conjectures on rational points and adelic mixing., Annales scientifiques de l'ENS, vol 41, 2008, p 383- 435. (arXiv). (7) Arithmetical and geometrical aspects of homogeneous diophantine approxi- mation by algebraic numbers in a given number field, Proceedings "Dynamical systems and Diophantine Approximations", S´eminaireset congr`esvol 22, S.M.F. (8) A non-homogeneous orbit closure of a diagonal subgroup, Annals of Maths, 171 (1), 2010, p557-570. (arXiv). (9) With Y. Coud`ene, Horocycles r´ecurrents sur des surfaces de volume infini, Geometriae Dedicata, Volume 149.1 (2010), p 231-242. (article) (10) Appendix of an article of S. Lim et H. Oh, On the distribution of orbits of geometrically finite hyperbolic groups on the boundary, Ergod. Th. and Dynam. Sys., Vol 32 (2012), 173-189. ( arXiv). (11) With B. Weiss, Lattice action on the plane revisited, Geometriae Dedicata, Vol 157.1 (2012), p1-21. (Lien arXiv). (12) With B. Schapira, Distribution of orbits in R2 of a finitely generated group of SL(2; R), American Journal of Mathematics(ArXiv). (13) With S. Gou¨ezelet F. Math´eus, Sharp lower bounds for the asymptotic en- tropy of symmetric random walks, to appear in Groups, Geometry and Dynamics. (ArXiv). In preparation (14) With Yves Guivarc'h, Linear random walks on the plane. (15) With Barbara Schapira, Nondivergent unipotent orbits for convex-concompact hyperbolic manifolds (16) With S. Gou¨ezelet F. Math´eus, Entropy and drift in word-hyperbolic groups Remerciements Je remercie chaleureusement Nalini Anantharaman, Amos Nevo et Arnaldo Nogueira (N.A., A.N. et A.N.) d'avoir accept´ede rapporter cette HDR. C'est un travail ingrat, et je leur en suis particuli`erement reconnaissant. Nalini, Yves Benoist, Fran¸coiseDal'bo, Marc Peign´eet Bertrand R´emy ont accept´ede faire partie de mon jury : c'est pour moi un grand honneur. Un grand Merci `aeux. Ce travail n'aurait bien s^urpas ´et´epossible sans mes collaborateurs Barak Weiss, Barbara Schapira et Yves Guivarc'h; il est beaucoup plus motivant de travailler `aplusieurs. Ce m´emoirene concerne qu'une partie de mes travaux, et je suis ´egalement d´ebiteurde mes collaborateurs sur d'autres sujets, Yves Coud`ene, Alex Gorodnik, Hee Oh, S´ebastienGou¨ezelet Fr´ed´ericMath´eus.Remerciements sp´eciaux`aBarbara et S´ebastien, qui ont relu quelques passages de ce m´emoire et m'ont fait part de leurs commentaires, ce qui m'a beaucoup aid´e`aam´eliorer le texte. C'est un plaisir quotidien que de faire partie de l'´equipe de th´eorieergodique de l'IRMAR, qui est un environnement tr`esenrichissant math´ematiquement, mais aussi tr`esagr´eablehumainement. Merci en particulier `aJean-Pierre, St´ephane, Serge, Ludo, Ania, Dimitri, Vincent et Juan. Enfin, un gros merci `aAxelle. 7 Introduction his HDR 2 is constructed around three papers, and a work-in-progress. These T four papers represent a large part of my work after the PhD, but not every- thing; this choice is motivated by the fact that all four can be considered partial answers to the same question. My goal here is not to reproduce those articles, but rather to put them into the same context. I tried to explain the main ideas without diving (too much) into squalid details. Overall, I tried to write something easy to read but that gives a precise idea of the actual technical problems and the tools involved to deal with them. For more precise statements, one can always refer to the original article, but hopefully that will not be necessary. Also, this is an HDR and not a review article; it focuses on my own work together with my coauthors Barak, Barbara and Yves. This means that very relevant results of others around this topic will often only be alluded to. In some cases, my results have been later improved by others (in chapter 1) but it's the 'old' result that is discussed, also because it is often simpler to state and under- stand. The initial (vague) question is: Take a (discrete) subgroup Γ ⊂ SL(2; R), a nonzero vector u 2 R2 and choose an element γ 2 Γ at random. What is γu, where is it on the plane ? My belief is that this question is not that interesting, but that the methods to answer it are. The typical objective in this memoir (at least in the first three chapters) will be to find asymptotic formulas for expressions of the form X γu f ; T α γ2Γ;kγk≤T 2short for 'Habilitation `aDiriger des Recherches', that is, the present text. 9 10 INTRODUCTION where α is a parameter, and f a continuous, compactly supported function on R2. Chapter 1. The first chapter, based on [31] - (5) in the publication list -, de- scribes the 'large scale' answer for lattices, through the analysis of homogeneous limits of Haar measure in representations. The link with the initial problem is the following: assume Γ = SL(2; R) (it's not discrete, but the question can be asked anyway) and that 'randomly' means taken with respect to Haar measure, restricted to a ball of radius T > 0, and normalized to be a probability. To un- derstand the limit behavior of this probability measure, first apply an homothety of ratio 1=T , so the probability is supported in a ball of radius 1 and we can hope (and prove) some convergence. Once we have found a limit in the space of matrices, we can use the evaluation map M 7! Mu (which commutes indeed with homotheties) to compute the limit distribution of vectors. One can do exactly same thing for any semisimple, connected, non-compact linear Lie group. A very practical problem arise in this context: what is the normalization required on the Haar measure restricted to a ball a radius T , that is, what is the asymptotic growth of the Haar measure ? I explain the procedure to find it and illustrate it on the automorphism representation of the split form of the exceptional Lie group of type G2.